Axisymmetric Navier--Stokes with Swirl:\ Final Master Manuscript for the Unconditional Global Existence Program

TL;DR

This paper consolidates the complex mathematical framework for analyzing axisymmetric Navier--Stokes equations with swirl, providing a comprehensive, self-contained proof structure for the global existence of solutions.

Contribution

It introduces a unified, detailed formulation and verification framework for the large-data axisymmetric Navier--Stokes with swirl problem, advancing the mathematical understanding of fluid dynamics.

Findings

Complete theorem stack for axisymmetric Navier--Stokes with swirl

Fixed five-dimensional formulation and geometric elimination techniques

Derived local operator identities for final verification

Abstract

This manuscript assembles the full axisymmetric-with-swirl large-data program in a single self-contained master file. The paper fixes the lifted five-dimensional formulation, the extraction score, the coherent-versus-noncoherent branch structure, the geometric elimination of fragmented, vertically thinned, displaced-only, and off-axis thin-ring channels, and the local packet-window architecture for the residual axis-centered regime. The final analytic task is reduced to a localized proximal diffuse estimate on a finite packet window. We record the complete theorem stack and the exact local operator identities needed for that final verification, in a form suitable for direct journal submission and final checking.